∫ −512000000−2048000000x2−3840000x3−2048000000x4−10240000x5−5120000x7+(2048000000x2+4096000000x4) log(x)_____________________________________________________________________________________ x10+1200x7 log(x)+480000x4 log2(x)+64000000x log3(x) dx

3.72.12 \(\int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+(2048000000 x^2+4096000000 x^4) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {\left (2+4 x^2\right )^2}{\left (\frac {x^3}{400}+\log (x)\right )^2} \]

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Rubi [F] time = 1.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-512000000 - 2048000000*x^2 - 3840000*x^3 - 2048000000*x^4 - 10240000*x^5 - 5120000*x^7 + (2048000000*x^2

+ 4096000000*x^4)*Log[x])/(x^10 + 1200*x^7*Log[x] + 480000*x^4*Log[x]^2 + 64000000*x*Log[x]^3),x]

[Out]

-512000000*Defer[Int][1/(x*(x^3 + 400*Log[x])^3), x] - 2048000000*Defer[Int][x/(x^3 + 400*Log[x])^3, x] - 3840

000*Defer[Int][x^2/(x^3 + 400*Log[x])^3, x] - 2048000000*Defer[Int][x^3/(x^3 + 400*Log[x])^3, x] - 15360000*De

fer[Int][x^4/(x^3 + 400*Log[x])^3, x] - 15360000*Defer[Int][x^6/(x^3 + 400*Log[x])^3, x] + 5120000*Defer[Int][

x/(x^3 + 400*Log[x])^2, x] + 10240000*Defer[Int][x^3/(x^3 + 400*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1280000 \left (1+2 x^2\right ) \left (-400-800 x^2-3 x^3-2 x^5+1600 x^2 \log (x)\right )}{x \left (x^3+400 \log (x)\right )^3} \, dx\\ &=1280000 \int \frac {\left (1+2 x^2\right ) \left (-400-800 x^2-3 x^3-2 x^5+1600 x^2 \log (x)\right )}{x \left (x^3+400 \log (x)\right )^3} \, dx\\ &=1280000 \int \left (-\frac {\left (1+2 x^2\right )^2 \left (400+3 x^3\right )}{x \left (x^3+400 \log (x)\right )^3}+\frac {4 x \left (1+2 x^2\right )}{\left (x^3+400 \log (x)\right )^2}\right ) \, dx\\ &=-\left (1280000 \int \frac {\left (1+2 x^2\right )^2 \left (400+3 x^3\right )}{x \left (x^3+400 \log (x)\right )^3} \, dx\right )+5120000 \int \frac {x \left (1+2 x^2\right )}{\left (x^3+400 \log (x)\right )^2} \, dx\\ &=-\left (1280000 \int \left (\frac {400}{x \left (x^3+400 \log (x)\right )^3}+\frac {1600 x}{\left (x^3+400 \log (x)\right )^3}+\frac {3 x^2}{\left (x^3+400 \log (x)\right )^3}+\frac {1600 x^3}{\left (x^3+400 \log (x)\right )^3}+\frac {12 x^4}{\left (x^3+400 \log (x)\right )^3}+\frac {12 x^6}{\left (x^3+400 \log (x)\right )^3}\right ) \, dx\right )+5120000 \int \left (\frac {x}{\left (x^3+400 \log (x)\right )^2}+\frac {2 x^3}{\left (x^3+400 \log (x)\right )^2}\right ) \, dx\\ &=-\left (3840000 \int \frac {x^2}{\left (x^3+400 \log (x)\right )^3} \, dx\right )+5120000 \int \frac {x}{\left (x^3+400 \log (x)\right )^2} \, dx+10240000 \int \frac {x^3}{\left (x^3+400 \log (x)\right )^2} \, dx-15360000 \int \frac {x^4}{\left (x^3+400 \log (x)\right )^3} \, dx-15360000 \int \frac {x^6}{\left (x^3+400 \log (x)\right )^3} \, dx-512000000 \int \frac {1}{x \left (x^3+400 \log (x)\right )^3} \, dx-2048000000 \int \frac {x}{\left (x^3+400 \log (x)\right )^3} \, dx-2048000000 \int \frac {x^3}{\left (x^3+400 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.34, size = 21, normalized size = 0.95 \begin {gather*} \frac {640000 \left (1+2 x^2\right )^2}{\left (x^3+400 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-512000000 - 2048000000*x^2 - 3840000*x^3 - 2048000000*x^4 - 10240000*x^5 - 5120000*x^7 + (20480000

00*x^2 + 4096000000*x^4)*Log[x])/(x^10 + 1200*x^7*Log[x] + 480000*x^4*Log[x]^2 + 64000000*x*Log[x]^3),x]

[Out]

(640000*(1 + 2*x^2)^2)/(x^3 + 400*Log[x])^2

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fricas [A] time = 0.67, size = 33, normalized size = 1.50 \begin {gather*} \frac {640000 \, {\left (4 \, x^{4} + 4 \, x^{2} + 1\right )}}{x^{6} + 800 \, x^{3} \log \relax (x) + 160000 \, \log \relax (x)^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5-2048000000*x^4-3840000*x^3-20480000

00*x^2-512000000)/(64000000*x*log(x)^3+480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm="fricas")

[Out]

640000*(4*x^4 + 4*x^2 + 1)/(x^6 + 800*x^3*log(x) + 160000*log(x)^2)

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giac [B] time = 0.15, size = 71, normalized size = 3.23 \begin {gather*} \frac {640000 \, {\left (12 \, x^{7} + 12 \, x^{5} + 1600 \, x^{4} + 3 \, x^{3} + 1600 \, x^{2} + 400\right )}}{3 \, x^{9} + 2400 \, x^{6} \log \relax (x) + 400 \, x^{6} + 480000 \, x^{3} \log \relax (x)^{2} + 320000 \, x^{3} \log \relax (x) + 64000000 \, \log \relax (x)^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5-2048000000*x^4-3840000*x^3-20480000

00*x^2-512000000)/(64000000*x*log(x)^3+480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm="giac")

[Out]

640000*(12*x^7 + 12*x^5 + 1600*x^4 + 3*x^3 + 1600*x^2 + 400)/(3*x^9 + 2400*x^6*log(x) + 400*x^6 + 480000*x^3*l

og(x)^2 + 320000*x^3*log(x) + 64000000*log(x)^2)

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maple [A] time = 0.02, size = 25, normalized size = 1.14


method	result	size

risch	\(\frac {2560000 x^{4}+2560000 x^{2}+640000}{\left (x^{3}+400 \ln \relax (x )\right )^{2}}\)	\(25\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((4096000000*x^4+2048000000*x^2)*ln(x)-5120000*x^7-10240000*x^5-2048000000*x^4-3840000*x^3-2048000000*x^2-

512000000)/(64000000*x*ln(x)^3+480000*x^4*ln(x)^2+1200*x^7*ln(x)+x^10),x,method=_RETURNVERBOSE)

[Out]

640000*(4*x^4+4*x^2+1)/(x^3+400*ln(x))^2

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maxima [A] time = 0.40, size = 33, normalized size = 1.50 \begin {gather*} \frac {640000 \, {\left (4 \, x^{4} + 4 \, x^{2} + 1\right )}}{x^{6} + 800 \, x^{3} \log \relax (x) + 160000 \, \log \relax (x)^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5-2048000000*x^4-3840000*x^3-20480000

00*x^2-512000000)/(64000000*x*log(x)^3+480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm="maxima")

[Out]

640000*(4*x^4 + 4*x^2 + 1)/(x^6 + 800*x^3*log(x) + 160000*log(x)^2)

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mupad [B] time = 4.40, size = 21, normalized size = 0.95 \begin {gather*} \frac {640000\,{\left (2\,x^2+1\right )}^2}{{\left (400\,\ln \relax (x)+x^3\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(2048000000*x^2 - log(x)*(2048000000*x^2 + 4096000000*x^4) + 3840000*x^3 + 2048000000*x^4 + 10240000*x^5

+ 5120000*x^7 + 512000000)/(64000000*x*log(x)^3 + 1200*x^7*log(x) + 480000*x^4*log(x)^2 + x^10),x)

[Out]

(640000*(2*x^2 + 1)^2)/(400*log(x) + x^3)^2

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sympy [A] time = 0.15, size = 29, normalized size = 1.32 \begin {gather*} \frac {2560000 x^{4} + 2560000 x^{2} + 640000}{x^{6} + 800 x^{3} \log {\relax (x )} + 160000 \log {\relax (x )}^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4096000000*x**4+2048000000*x**2)*ln(x)-5120000*x**7-10240000*x**5-2048000000*x**4-3840000*x**3-204

8000000*x**2-512000000)/(64000000*x*ln(x)**3+480000*x**4*ln(x)**2+1200*x**7*ln(x)+x**10),x)

[Out]

(2560000*x**4 + 2560000*x**2 + 640000)/(x**6 + 800*x**3*log(x) + 160000*log(x)**2)

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